A subtree of a tree is any induced subgraph that is again a tree (i.e., connected). The mean subtree order of a tree is the average number of vertices of its subtrees. This invariant was first analyzed in the 1980s by Jamison. An intriguing open question raised by Jamison asks whether the maximum of the mean subtree order, given the order of the tree, is always attained by some caterpillar. While we do not completely resolve this conjecture, we find some evidence in its favor by proving different features of trees that attain the maximum. For example, we show that the diameter of a tree of order $n$ with maximum mean subtree order must be very close to $n$. Moreover, we show that the maximum mean subtree order is equal to $n-2{log}_{2}n+O\left(1\right)$. For the local mean subtree order, which is the average order of all subtrees containing a fixed vertex, we can be even more precise: we show that its maximum is always attained by a broom and that it is equal to $n-{log}_{2}n+O\left(1\right)$.

树的子树是任何再次是树（即，连接）的诱导子图。一棵树的平均子树顺序是其子树的平均顶点数。Jamison 在 1980 年代首次分析了这个不变量。Jamison 提出了一个有趣的开放性问题，询问在给定树的顺序的情况下，平均子树顺序的最大值是否总是由某些毛毛虫达到。虽然我们没有完全解决这个猜想，但我们通过证明达到最大值的树的不同特征，找到了一些支持它的证据。例如，我们证明一棵树的直径$n$ 最大平均子树顺序必须非常接近 $n$. 此外，我们表明最大平均子树顺序等于$n-2{日志}_{2}n+哦\left(1\right)$. 对于局部平均子树顺序，即包含固定顶点的所有子树的平均顺序，我们可以更精确：我们证明它的最大值总是由扫帚达到，并且它等于$n-{日志}_{2}n+哦\left(1\right)$.